The Mandelbrot set puts some geometry into the fundamental observation
above. The precise definition is: The Mandelbrot set M
consists of all of those (complex) c-values for which the
corresponding orbit of 0 under x2 + c
does not escape to infinity.
From our previous calculations, it follows that
c = 0, -1, -1.1, -1.3, -1.38, and i
all lie in the Mandelbrot set,
whereas c = 1 and c = 2i do not.

At this point, a natural question is: Why would anyone care about the
fate of the orbit of 0 under x2 + c ?
Why not the orbit of i ? Or
2 + 3i ? Or any other complex seeds, for that matter? As we will see
below, there is a very good reason for inquiring about the fate of the
orbit of 0; somehow the orbit of 0 tells us a tremendous amount about
the fate of other orbits under x2 + c.

Before turning to this idea, note that the very definition of the
Mandelbrot set gives us an algorithm for computing it. We simply
consider a square in the complex plane (usually centered at the origin
with sides of length 4). We overlay a grid of equally spaced points in this
square. Each of these points is to be considered a complex c-value.
Then, for each such c, we ask the computer to check whether the
corresponding orbit of 0 goes to infinity (escapes) or does not go to
infinity (remains bounded). In the former case, we leave the
corresponding c-value (pixel) white. In the latter case, we paint
the c-value black. Thus the black points in Figure 3
represent the Mandelbrot set.

Figure 3. The Mandelbrot set

Two points need to be made. Figure 3 is only an approximation
of the Mandelbrot set. Indeed, it is not possible to determine whether
certain c-values lie in the Mandelbrot set. We can only iterate a
finite number of times to determine if a point lies in M .
Certain c-values close to the boundary of M have orbits
that escape only after a very large number of iterations.

A second question is: How do we know that the orbit of 0 under x2 + c
really does escape to infinity? Fortunately, there is an easy
criterion which helps:

The Escape Criterion: Suppose |c| is less than or equal to 2.
If the orbit of 0
under x2 + c ever lands outside of the circle of radius 2 centered at
the origin, then this orbit definitely tends to infinity.

It may seem that this criterion is not too valuable, as it only works
when |c| is less than or equal to 2. However, it is known that the entire Mandelbrot set
lies inside this disk, so these are the only c-values we need
consider anyway.